What is the proportion of women with breast cancer in the population having a positive test?
Activity: Medical diagnosis
Use probability theory to calculate the exact probability of a women having breast cancer once we know she got a positive test.
Hint. Consider the following events:
\(A\) = “To have cancer”, \(A^c\) = “Not to have cancer” and
\(B\) = “Test is positive”.
Independence between events
Two events \(A\) and \(B\) are said to be independent, denoted \(A \perp B\), if
\[
\text{P}(A) = \text{P}(A|B).
\]
A practical equivalent definition is \(A\) and \(B\) are independent if
\[
\text{P}(A \cap B) = \text{P}(A) \text{P}(B).
\]
Conditional independence
Two events \(A\) and \(B\) are said to be conditionally independent given \(C\), denoted \(A \perp B | C\), if
\[
\text{P}(A|C) = \text{P}(A|B \cap C),
\]
or
\[
\text{P}(A \cap B | C) = \text{P}(A |C)\text{P}(B |C).
\]
Activity
Check if event A = “face is even” and event B = “the face is smaller or equal than 4” are independent when rolling a 6-faced dice.
Check if event A = “face is 6 in the blue dice” and B = “face is 6 in the red dice” are independent when rolling a blue and red 6-faced dice.
Check if event A = “face is 6 in the blue dice” and B = “face is 6 in the red dice” are conditionally independent when rolling a blue and red 6-faced dice when we know about event C = “the sum is even”.
Activity: Monty Hall problem
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Random variables
A random variable (r.v.) is a function from a sample space to a measurable space, like the real numbers.
We can thing of a random variable as a variable taking values randomly.
If values are discrete, the function assigning a probability to its values is called probability mass function (pmf).
Discrete random variables
Discrete r.v. take values in a countable set of elements. For example a finite set of numbers or the integers.
Discrete r.v. can be characterised with giving \(\text{P}(X = x)\) for each possible value \(x\). Function \(p(x) = \text{P}(X = x)\) is called the probability mass function (pmf).
Example: Discrete r.v. \(X\) is uniquely determined by
\(X \in \{1, 2, 3\}\) and
the pmf: \(p(1) = 0.25\), \(p(2) = 0.5\) and \(p(3) = 0.25\).
Continuous random variables
Continuous random variables take values in the real line.
Possible events for continuous r.v. are combinations of subsets the real line.
Continuous r.v. are characterised by providing \(\text{P}(X \in E)\) for any event \(E\).
The probability density function (pdf) of a coninuous r.v. is a function \(f(x)\) such that ((X E) = _{x E} f(x) dx. )
Example: Continuous r.v. \(X\) is uniquely determined with the pdf \(f(x)\) defined as \(f(x) = \frac{1}{3}\) if \(x \in (0,3)\) and \(f(x) = 0\) otherwise.
The expected value or mean of a r.v.
The expected value of a r.v. \(X\), denoted \(\mathbb{E}[X]\), is the mean value we expect after an infinite number of runs.
For a discrete r.v. \(X\), the expected value is \[
\mathbb{E}[X] = \sum_{x} x \;p(x) \;\;\;\;\;\;\;\;\;\;[\sum_{x} x \;\text{P}(X = x)]
\]
For a continuous r.v. \(X\), the expected value is \[
\mathbb{E}[X] = \int_{-\infty}^{\infty} x \;f(x) dx
\]
Example: the expected value of the r.v. with the value of rolling a 6-faced dice is
For any function \(g(x)\), we can defined the expected value of \(g(X)\) as
\(\mathbb{E}[g(X)] = \sum_{x} g(x) \;p(x)\) when \(X\) is discrete or
\(\mathbb{E}[g(X)] = \int_{-\infty}^{\infty} g(x) \;f(x) dx\) when continuous.
Some common probability distributions
Categorical distribution
The categorical distribution is the distribution obtained after associating natural number \(1\) to \(k\) to a set of events with a certain probability \(\boldsymbol\pi = (\pi_1, \dots, \pi_k)\).
For a categorical r.v. \(X\), denoted \(X \sim Cat(\boldsymbol\pi = (\pi_1, \dots, \pi_k))\),
\(\mu = \mathbb{E}[X] = \sum_{i \leq k} i\; \pi_k\) and
\(\text{var}[X] = \sum_{i \leq k} (i - \mu)^2\; \pi_k\).
Gaussian (normal) distribution
The gaussian (normal) distribution is the distribution that reduces the probability exponentially (with velocity \(1/\sigma^2\)) for events far from a certain center (\(\mu\)).
For a gaussian r.v. \(X\), \(X \sim N(\mu, \sigma)\),
\(\mathbb{E}[X] = \mu\) and
\(\text{var}[X] = \sigma^2\).
Binomial distribution
The binomial distribution counts the number of success after repeating a certain experiments \(n\) times when the probability of success is \(\pi\).
For a binomial r.v. \(X\), \(X \sim Bin(n, \pi)\),
\(\mathbb{E}[X] = n \pi\) and
\(\text{var}[X] = n \pi (1-\pi)\).
Poisson distribution
The Poisson distribution counts the number of success detected in an interval of time or space when the expected number of successes is \(\lambda\).
For a Poisson r.v. \(X\), \(X \sim Poiss(\lambda)\),
Thing about possible variables that can be modelled with one of the four seen distributions:
Categorical distribution
Gaussian distribution
Binomial distribution
Poisson distribution
Think about possible parameters to use.
Joint probability distribution
The joint probability distribution is a multivariate model for random variables. The joint probability distribution of r.v. \(X_1, \dots, X_k\) is completely determined by providing
the probability of all possible events for discrete r.v.’s
If we transform a r.v. with a function \(g(x)\) we get another r.v.
Linear transformations. If we linearly transform a r.v. \(\boldsymbol X=(X_1,\dots,X_k)\) with \(f(\boldsymbol x) = \boldsymbol A \boldsymbol x + \boldsymbol b\) then
\(\mathbb{E}[f(\boldsymbol X)] = \boldsymbol A \boldsymbol\mu + \boldsymbol b\) where \(\boldsymbol\mu = \mathbb{E}[\boldsymbol X]\), and
\(\text{cov}[f(\boldsymbol X)] = \boldsymbol A \boldsymbol\Sigma \boldsymbol A^{T}\) where \(\boldsymbol\Sigma = \text{cov}[\boldsymbol X]\).
For non-linear transformations we can simulate the r.v. to obtain these expected values.
Example: The log-normal distribution is defined as the distribution obtained after exponentiating a gaussian r.v., \(f(x) = e^x\), \(Y = e^X\), where \(X \sim N(\mu,\sigma)\). Check using simulating gaussian r.v. that
If \(X_1, \dots, X_n\) are r.v. equally distributed with expected value \(\mathbb{E}[X]\) and variance \(\text{var}[X]\). Then, when \(n\) is big, we have
Next plot shows the distribution of 1000 realisations of a r.v. obtained as the average of 30 binomial distributions with parameters \(n=10\) and \(\pi=0.25\). Blue line represents the gaussian distribution with \(\mu=n \pi\) and \(\sigma^2=\sqrt{n \pi (1-\pi) / 30}\)
The central limit theorem
If \(X_1, \dots, X_n\) are r.v. equally distributed with expected value \(\mathbb{E}[X]\) and variance \(\text{var}[X]\). Then, when \(n\) is big, we have
Next plot shows the distribution of 1000 standardised realisations of a r.v. obtained as the average of 30 binomial distributions with parameters \(n=10\) and \(\pi=0.25\). Blue line represents the gaussian distribution with \(\mu=0\) and \(\sigma^2=1\)
Sampling distribution
If \(x_1, \dots, x_n\) is a sample i.i.d with expected value \(\mathbb{E}[X]\). Then, when \(n\) is big, we have
Next plot shows the distribution of 1000 standardised realisations of a r.v. obtained as the average of 30 binomial distributions with parameters \(n=10\) and \(\pi=0.25\). Normal and Student distributionss are compared.
Final activity
Fix some r.v. \(Y\) after some transformation \(f(x)\).
For example by transforming certain binomial distribution with function \(f(x) = \sin(x)\).
Take a sample of size \(50\) and calculate the mean of this sample. We we call it: \(x_1\).
Repeat the process 1000 times obtaining a sample of means: \(X_m = \{x_1, \dots, x_{1000}\}\).
Visualise the resulting distribution of sample \(X_m\).
Think about how we can use \(x_i\) to obtain information of \(\mathbb{E}[Y]\)?
That’s all for today
Next week session
Overview of probability and statistics
Chi-squared test: goodness of fit, independence
Student’s t-test: one sample, paired samples, independent two-sample
